THE SILVER BAR

A SILVER PROSPECTOR was unable to pay his March rent in advance. He owned a bar of pure silver, 31 inches long, so he made the following arrangement with his landlady. He would cut the bar, he said, into smaller pieces. On the first day of March he would give the lady an inch of the bar, and on each succeeding day, he would add another inch to her amount of silver. She would keep this silver as security. At the end of the month, when the prospector expected to be able to pay his rent in full, she would return the pieces to him. March has 31 days so one way to cut the bar would be to cut it into 31 sections, each an inch long. But since it required considerable labor to cut the bar, the prospector wished to carry out his agreement with the fewest possible number of pieces. For example,
he might give the lady an inch on the first day, another inch the second day, then on the third day he
could take back the two pieces and give her a solid 3-inch section.

Assuming that portions of the bar are traded back and forth in this fashion, see if you can determine the smallest number of pieces into which the prospector needs to cut his silver bar.

SOLUTION

The prospector can keep his agreement by cutting his 3I-inch silver bar into as few as five sections with lengths of 1, 2, 4, 8, and 16 inches. On the first day he gives the landlady the I-inch piece, the next day he takes it back and gives her the 2-inch piece, the third the day he gives her the I-inch piece again, the fourth day he takes back both pieces and gives her the 4-inch piece. By giving and trading in this manner, he can add an inch to her amount each day for the full month of 31 days.

The solution to this problem can be expressed very neatly in the binary system of arithmetic. This is amethod of expressing integers by using only the digits 1 and O. In recent years it has become an important system because most giant electronic computers operate on a binary basis. Here is how the number 27, for example, would be written if we are using the binary system:

11011

How do we know that this is 27? The way to translate it into our decimal system is as follows. Above the digit on the extreme right of the binary number, we write "1." Above the next digit, moving left, we write "2"; above the third digit from the left, we write "4"; above the next digit, "8"; and above the last digit on the left, "16." (See the illustration.) These values for the series 1, 2, 4, 8, 16, 32 . . . , in which each number is twice the preceding one. 16 8 4 2 1

11011

The next step is to add together all the values that are above us in the binary number. In this case, the
values are 1, 2, 8, 16 (4 is not included because it is above a 0). They add up to 27, so the binary number 11011 is the same as 27 in our number system. Any number from 1 to 31 can be expressed in this way with a binary number of no more than five digits. In exactly the same way, any number of inches of silver from 1 to 31 can be formed with five pieces of silver if the lengths of the five pieces are 1, 2, 4, 8, and 16 inches.

The table here lists the binary numbers for each day in March. You will note that on March 27 the number is 11011. This tells us that the landlady's 27 inches of silver will consist of the I-inch, 2-inch, 8-inch, and 16-inch sections. Pick a day at random and see how quickly you can learn from the chart exactly which pieces of silver will add to an amount that corresponds to the number of the day.